# Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Apriori energy estimate for strange PDE: $\partial_{tt} \partial_{yy} u + \partial_{xx} u = F$

I have this rather obscure-looking (weak-form of a) PDE $$(\partial_y u'', \partial_y v)+(\partial_x u, \partial_x v) = (f,\partial_x v)+(g,v)\quad\forall v\in \mathcal{D}.$$ Here $(\cdot,\cdot)$ is ...
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### Log-convexity of a generalisation of weigthed-Lp norms

Log-convexity of weigted Lp norms Let the weighted norm of a positive vector $q\in \mathrm R_+^d$ with weights $a\in \mathrm R_+^d$, be defined as $${||q||}_{(p,a)} = (\sum_i a_i q_i^{p})^{1/p}$$ Then ...
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### Gateaux differentiability of squared $\ell_p$ norm

Consider the Euclidean vector space $\mathbb{R}^n$ and the squared $p$-norm $||\cdot||_p^2$ for ($1<p<\infty$). I'm trying to understand if $||\cdot||_p^2$ is Gateaux differentiable at any point ...
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### a new norm on piecewise linear functions

In symplectic geometry, there is a norm/metric defined on (time dependent) Hamiltonian functions $H:M\times [0,1]\to \mathbb{R}$ which is quite useful : A sufficient condition for this to make sense ...
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### Must $V$ have codimension 1 in a dense subspace if its closure is codimension 1 in the whole space?

Let $H$ be a (real) Hilbert space and $V\subset U\subset H$ where $U$ is dense proper subspace of $H$ and $V$ is a proper subspace of $U$. It is given that the closure of $V$, $\overline{V}$ has ...
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### Show that $S:c_0\to c_0$ given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$ is an isometry for the $l_\infty$ norm

Let $S:c_{0}\to c_{0}$ be given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$. Show that $S$ is an isometry, where $c_0$ is the subspace of $l_{\infty}$ consisting of sequences which converge to $0$. My ...
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### Which metrics (on vector spaces) can be induced?

Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$. I ...
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### Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.

Problem: Show that if $\mathbb{L}(\mathbb{R}^2)$, the space of the linear functionals in $\mathbb{R}^2$, with the matrix norm $\|\cdot\|_p$ ($p > 1$) is isomorphic with the $\mathbb{R}^2$ space ...
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### Prove that $BE^{\alpha}$ is a Banach Space

Let $E$ a Banach space, can we prove that the set $$BE^{\alpha}=\{f \in S': \ \sup_{t>0}t^{\alpha}\|G(t)f\|_{E} < +\infty\}$$ is also a Banach space? Here $S'$ is the set of tempered ...
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### Norm on the space of vector fields

Let $M$ be a finite dimensional manifold. There exists the notion of the norm of a tangent vector field on M? In other words, the space of tangent vector fields is a normed space? Thank you in advance ...
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### Spanning set of support functionals in dual space

I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries: Let $X$ be a normed space and $X^*$ be the ...
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### The space $C_c$ of real-valued compactly supported, continuous functions is not a Banach space under any norm

This answer showed the space $c_{00}$ of compactly supported sequences, is not a Banach space under any norm. I wonder if the same is true for the space $C_c$ of real-valued compactly supported, ...
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### (Copy) Set of linear functionals span the dual space iff intersection of their kernels is {0} .

I have fully understood the following question and got a motivation from it. Set of linear functionals span the dual space iff intersection of their kernels is $\{0\}$. My question is what will be the ...
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Norm: A norm on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb R$ which assigns each vector $x$ its length $\|x\| \in \mathbb R,$ such that for all $\lambda \in R$ and $x, y \in V$ the ...